Measure-theoretically mixing subshifts of minimal word complexity

We resolve a long-standing open question on the relationship between measure-theoretic dynamical complexity and symbolic complexity by establishing the exact word complexity at which measure-theoretic strong mixing manifests: For every superlinear f : N → N f : \mathbb {N} \to \mathbb {N} , i.e., f ( q ) q → ∞ \frac {f(q)}{q} \to \infty , there exists a subshift admitting a (strongly) mixing of all orders probability measure with word complexity p p such that p ( q ) f ( q ) → 0 \frac {p(q)}{f(q)} \to 0 . For a subshift with non-superlinear word complexity p p , lim inf p ( q ) q > ∞ \liminf \frac {p(q)}{q} > \infty , every ergodic probability measure is partially rigid.